Uniqueness of quasi-Einstein metrics on 3-dimensional homogeneous manifolds

Abstract

The purpose of this article is to study the existence and uniqueness of quasi-Einstein structures on 3-dimensional homogeneous Riemannian manifolds. To this end, we use the eight model geometries for 3-dimensional manifolds identified by Thurston. First, we present here a complete description of quasi-Einstein metrics on 3-dimensional homogeneous manifolds with isometry group of dimension 4. In addition, we shall show the absence of such gradient structure on Sol 3 , which has 3-dimensional isometry group. Moreover, we prove that Berger's spheres carry a non-trivial quasi-Einstein structure with non gradient associated vector field, this shows that a theorem due to Perelman can not be extend to quasi-Einstein metrics. Finally, we prove that a 3-dimensional homogeneous manifold carrying a gradient quasi-Einstein structure is either Einstein or H 2 κ × R.

Non-trivial m-quasi-Einstein metrics on simple Lie groups

“…The above discussion shows that not every m-quasi-Einstein metric on a compact manifold with λ 0 is trivial. The same conclusion can be deduced from counterexamples constructed in [5]. For example, the Berger sphere admits metrics where λ is positive, negative or zero.…”

Three dimensional m‐quasi Einstein manifolds with degenerate Ricci tensor

“…We note that not much is known about three dimensional (denoted by 3‐d, below) m ‐QE manifolds, though some characterizations were known in certain cases, for instance when is locally conformally flat as mentioned above, or homogeneous .…”

Ricci solitons on almost Kenmotsu 3-manifolds

“…4/ R is a rigid gradient Ricci soliton (see [20,21]). For other results on existence of quasi-Einstein metrics on this product we refer the reader to [22].…”

“…Finally, since is a positive constant, using the previous relation, (23) and (24) we have a D 0. In this context, (22) …”